Suppose f: D --> R is a function whose domain contains [a,b] for some a,b $\displaystyle \in$ R amd f is integrable on [a,b]. suppose a < c < b.

1. Suppose A is a set of real number that is bounded above. Prove that for any $\displaystyle \epsilon$ > 0 there is a number a $\displaystyle \in$A Such that supA-a < $\displaystyle \epsilon$.

(Hint: Prove by contradiction. If there is an $\displaystyle \epsilon$ >0 such that supA-a >= $\displaystyle \epsilon$ for all a $\displaystyle \in$ A then what does this say about the number supA - $\displaystyle \epsilon$ ?)